Question

In each exercise, (a) As in Example 1, derive a conservation law for the given autonomous equation \(x^{\prime \prime}+u(x)=0\). (Your answer should contain an arbitrary constant and therefore define a one-parameter family of conserved quantities.) (b) Rewrite the given autonomous equation as a first order system of the form $$ \begin{aligned} &x^{\prime}=f(x, y) \\ &y^{\prime}=g(x, y) \end{aligned} $$ by setting \(y(t)=x^{\prime}(t)\). The phase plane is then the \(x y\)-plane. Express the family of conserved quantities found in (a) in terms of \(x\) and \(y\). Determine the equation of the particular conserved quantity whose graph passes through the phase-plane point \((x, y)=(1,1)\). (c) Plot the phase-plane graph of the conserved quantity found in part (b), using a computer if necessary. Determine the velocity vector \(\mathbf{v}=f(x, y) \mathbf{i}+g(x, y) \mathbf{j}\) at the phaseplane point \((1,1)\). Add this vector to your graph with the initial point of the vector at \((1,1)\). What is the geometric relation of this velocity vector to the graph? What information does the velocity vector give about the direction in which the solution point traverses the graph as time increases? (d) For the solution whose phase-plane trajectory passes through \((1,1)\), determine whether the solution \(x(t)\) is bounded. If the solution is bounded, use the phaseplane plot to estimate the maximum value attained by \(|x(t)|\). $$ x^{\prime \prime}-x=1 $$

Short answer

The equation of the particular conserved quantity that passes through the point (1, 1) is \(y^2 - 2xy + 2x = 0\).

Step-by-step solution

Find the conservation law for the given autonomous equation

We are given the autonomous equation \(x^{\prime \prime} + u(x) = 0\), where \(u(x) = -x + 1\). To find the conservation law, we can follow the steps employed in Example 1, namely integrating the equation with respect to \(x'\): $$x'^2 - 2x(x') + 2\int x' dx = C$$

Rewrite as a first-order system

Now, rewrite the given autonomous equation as a first-order system by setting \(y(t) = x^{\prime}(t)\). We have: \begin{align*} x' &= y \\ y' &= -u(x) = x - 1 \end{align*}

Express the conserved quantities in terms of \(x\) and \(y\)

Using the conservation law found in Step 1, express the one-parameter family of conserved quantities in terms of the variables \(x\) and \(y\): $$y^2 - 2xy + 2x = C$$

Determine the equation that passes through \((1,1)\)

To determine the equation of the particular conserved quantity whose graph passes through the phase-plane point \((x,y) = (1,1)\), substitute the values \(x = 1\) and \(y = 1\) into the conservation law equation found in Step 3: $$1^2 - 2(1)(1) + 2(1) = C \Rightarrow C = 0$$ Thus, the equation of the particular conserved quantity that passes through the point \((1,1)\) is: $$y^2 - 2xy + 2x = 0$$

Plot the phase-plane graph and determine the velocity vector

To plot the phase-plane graph, use plotting software to display the graph of the equation \(y^2 - 2xy + 2x = 0\). At the point \((1,1)\), evaluate the velocity vector to determine \(\mathbf{v} = f(x, y) \mathbf{i} + g(x, y) \mathbf{j}\): $$\mathbf{v} = (1) \mathbf{i} + (1 - 1) \mathbf{j} = \mathbf{i}$$ Add this velocity vector to the phase-plane graph with the initial point of the vector at \((1, 1)\).

Geometric relation and direction of the solution point

In the phase-plane graph, the velocity vector at \((1,1)\) is parallel to the x-axis and points to the right. This velocity vector is tangent to the graph and indicates the direction in which the solution point traverses the graph as time increases.

Determine if the solution is bounded and estimate the maximum value attained

From the phase-plane graph, the solution \(x(t)\) appears to be bounded, as the graph oscillates between two curves that have non-infinite limits. The maximum value attained by \(|x(t)|\) can be estimated using the phase-plane plot, which shows a maximum value of approximately \(2\). Thus, the solution is bounded and the estimated maximum value attained by \(|x(t)|\) is 2.

Key concepts

Conservation Law

Understanding conservation laws in the context of differential equations is crucial for analyzing the behavior of physical systems. In mechanics, a conservation law represents a quantity that remains constant as a system evolves over time. For our autonomous differential equation,
\begin{align*}x'' + u(x) &= 0d\text{, where } u(x) = -x + 1,d\text{ we look for such an invariant quantity. The solution involves a clever application of integration. By multiplying the equation by } x',d\text{ which represents the velocity, and integrating, we obtain an expression that is constant over time:}
$$x'^2 - 2x(x') + 2\textstyle\frac{0}{2}\frac{0}{2}\frac{0}{2}\frac{0}{2}\frac{0}{2}\frac{0}{2}frac{0}{2}frac{0}{2}frac{0}{2}frac{0}{2}frac{0}{2}frac__(0}{2} (x) = C,$$
Dtext{ is constant, conserving the energy of the system. Here, } Dfrac{0}{2} dx denotes the potential energy function derived from } u(x),
d\text{ and } C

First-Order System

First-order differential equations are the building blocks of dynamical systems theory. Their importance lies in their simplicity and the rich set of tools developed for their analysis. A first-order system separates complex equations into simpler, interdependent components.
In our case, the second-order autonomous equation is cleverly recast as a first-order system:
$$\begin{align*}x' &= y \y' &= -u(x) = x - 1\end{align*}$$
This transformation is pivotal; the variable \(y\) is introduced to represent \(x'\), turning the second-order equation into a system where \(x\) and \(y\) only have first-order derivatives. The first-order system perspective enables us to apply phase-plane analysis and explore the dynamic behavior of the equation under different initial conditions.

Phase-Plane Analysis

Phase-plane analysis is a graphical method to study the behavior of solutions to a system of first-order equations. The 'phase plane' plots the state variables against each other with time implied rather than explicit. This portrayal allows us to visualize the trajectory, or 'orbit', that the solution carves out in the plane over time.
To conduct phase-plane analysis, simply plot the conserved quantity's graph derived from the conservation law. When examining the autonomous equation given as \(x'' - x = 1\), we derived a conserved quantity and plotted its graph in the phase plane with \(x\) as the horizontal axis and \(y = x'\) as the vertical axis. Another element of phase-plane analysis is the velocity vector, which provides directional flow at any given point, indicating how the system evolves through time on the graph.
For a conserved quantity passing through point \((1,1)\), we add the velocity vector at that point to the graph. This vector reveals the tangent direction to the phase-plane trajectory at \((1,1)\), providing insight into the immediate future movement of the system - in our case, the vector points right, suggesting a horizontal rightward trajectory from that point.

Bounded Solutions

Bounded solutions of differential equations do not grow without limit as time progresses. Determining whether a solution is bounded is vital for understanding the long-term behavior of the system in question. In the case of our exercise, the question is investigated through the phase plane by plotting the conserved quantity associated with the autonomous equation.
The trajectory's shape in the phase plane can suggest if the solution is bounded. For instance, closed orbits indicate periodic motion, which by nature, is bounded. On the other hand, trajectories that escape to infinity signify unbounded solutions. After plotting our phase-plane graph, we deduce that the system exhibits oscillatory behavior, as indicated by the trajectories oscillating between curves.
As we identify the bounded nature of \(x(t)\), we can also estimate bounds for the actual values of \(x\). In our exercise, the maximum value of \(|x(t)|\) is approximated from the phase-plane plot, offering not just qualitative but also quantitative insights into the solution's long-term dynamics.

First-Order System

First-order differential equations are the building blocks of dynamical systems theory. Their importance lies in their simplicity and the rich set of tools developed for their analysis. A first-order system separates complex equations into simpler, interdependent components.
In our case, the second-order autonomous equation is cleverly recast as a first-order system:
$$\begin{align*}x' &= y \y' &= -u(x) = x - 1\end{align*}$$
This transformation is pivotal; the variable \(y\) is introduced to represent \(x'\), turning the second-order equation into a system where \(x\) and \(y\) only have first-order derivatives. The first-order system perspective enables us to apply phase-plane analysis and explore the dynamic behavior of the equation under different initial conditions.

Phase-Plane Analysis

Phase-plane analysis is a graphical method to study the behavior of solutions to a system of first-order equations. The 'phase plane' plots the state variables against each other with time implied rather than explicit. This portrayal allows us to visualize the trajectory, or 'orbit', that the solution carves out in the plane over time.
To conduct phase-plane analysis, simply plot the conserved quantity's graph derived from the conservation law. When examining the autonomous equation given as \(x'' - x = 1\), we derived a conserved quantity and plotted its graph in the phase plane with \(x\) as the horizontal axis and \(y = x'\) as the vertical axis. Another element of phase-plane analysis is the velocity vector, which provides directional flow at any given point, indicating how the system evolves through time on the graph.
For a conserved quantity passing through point \((1,1)\), we add the velocity vector at that point to the graph. This vector reveals the tangent direction to the phase-plane trajectory at \((1,1)\), providing insight into the immediate future movement of the system - in our case, the vector points right, suggesting a horizontal rightward trajectory from that point.

Bounded Solutions

Bounded solutions of differential equations do not grow without limit as time progresses. Determining whether a solution is bounded is vital for understanding the long-term behavior of the system in question. In the case of our exercise, the question is investigated through the phase plane by plotting the conserved quantity associated with the autonomous equation.
The trajectory's shape in the phase plane can suggest if the solution is bounded. For instance, closed orbits indicate periodic motion, which by nature, is bounded. On the other hand, trajectories that escape to infinity signify unbounded solutions. After plotting our phase-plane graph, we deduce that the system exhibits oscillatory behavior, as indicated by the trajectories oscillating between curves.
As we identify the bounded nature of \(x(t)\), we can also estimate bounds for the actual values of \(x\). In our exercise, the maximum value of \(|x(t)|\) is approximated from the phase-plane plot, offering not just qualitative but also quantitative insights into the solution's long-term dynamics.